Simple and Compound Interest

Using the compound interest formula A = P(1 + r/100)^n, we have 1323 = 1200(1 + r/100)^2. Dividing by 1200 gives 1.1025 = (1 + r/100)^2. Taking the square root gives 1.05 = 1 + r/100, so r/100 = 0.05, meaning r = 5%.

Principal = 1000, Rate = 5% p.a. compounded half-yearly (2.5% per half-year), Time = 4 years (8 periods). Amount = 1000 * (1.025)^8 = 1000 * 1.2184 = 1218.4. Interest = 1218.4 - 1000 = 218.4, which rounds to 218.

Interest for 1st year = 25. Principal = 25 / 0.05 = 500. Interest for 2nd year = 500 * 0.05 = 25. Total interest after 2 years = 25 + 25 + (25 * 0.05) = 50 + 1.25 = 51.25.

For a period of one year, simple interest and compound interest are identical because interest is only calculated on the principal once. Therefore, the difference between them is 0.

Amount = P * (1 + r/100)^n = 2000 * (1 + 0.1)^2 = 2000 * 1.21 = 2420.

Simple interest for 2 years is 3000 * 0.06 * 2 = 360. Compound interest is 3000 * (1 + 0.06)^2 - 3000 = 3000 * 1.1236 - 3000 = 370.80. The difference is 370.80 - 360 = 10.80.

For 10% interest, the amount after 2 years is 1000 * (1.1)^2 = 1210. The compound interest is 1210 - 1000 = 210.

For (i) compounded annually: A = 75000 * (1 + 0.12)^1.5. For (ii) compounded half-yearly: A = 75000 * (1 + 0.06)^3. Calculating these gives approximately 88897 and 89326 respectively.

Half-yearly compounding gives a rate of 5% for each half-year. Over 1.5 years there are three half-years, so the amount is 12000 × 1.05^3 = Rs. 13891.50.

For 3 years, the difference between CI and SI is given by P * (r/100)^2 * (3 + r/100). Substituting r = 10, we get 992 = P * (0.1)^2 * (3.1) = P * 0.01 * 3.1 = P * 0.031. Solving for P gives 992 / 0.031 = 32000.

Principal = 12500, Rate = 8% per annum (2% per quarter), Time = 9 months (3 quarters). Amount = 12500 * (1 + 0.02)^3 = 12500 * 1.061208 = 13265.1. Interest = 13265.1 - 12500 = 765.1.

Scheme A earns 3600 at 12% simple interest for 2 years, so the amount invested in A is 15000. Thus, 20000 was invested in B, and its two-year compound interest at 10% is 20000(1.1^2 - 1) = 4200, so option B is correct.

At 10% per year compounded half-yearly, the rate per half-year is 5%. The compound interest on Rs. 50,000 for one year is 50,000 x (1.05^2 - 1) = Rs. 5,125.

Principal = 10800, Rate = 12.5% = 1/8. Amount = 10800 * (1 + 1/8)^3 = 10800 * (9/8)^3 = 10800 * 729 / 512 = 15377.34. Interest = 15377.34 - 10800 = 4577.34.